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VisionRes., Vol.36,No.14,pp.2141–2147,1996Couvright@1996ElsevierScienceLtd.Atl rightsreserved

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Binocular Alignment in Different Depth PlanesCASPER J. E]RKELENS,*TALEXANDRA J. M. MUIJS,* RAYMOND van EE*

Received18July 1995;in revisedform29 September1995

A generally accepted notion in binocular vision is that we see the world as if viewed by a single eye,the cyclopean eye. A consequence of seeing the world from a single point in space is that the outlinesof occluding and occluded surfaces have the same shape. We designed stereograms in whichsubjects aligned binocularly visible lines to each other. The lines were lying in different depthplanes. In the vicinity of occluded areas, binocular alignment was achieved by alignment of the linesin the eye that viewed the monocularly visible details. Stereograms in which shapes of surfaces lyingin different depth planes were compared to each other show that occluding and occluded surfacesdo not have the same shape: a square surface occludes rectangular surfaces in other depth planes ofwhich the horizontal widths are smaller than the vertical widths. This difference in perceived shapeis not possible if the centre of binocular direction has a fixed position in the head. Copyright 01996Elsevier Science Ltd.

Binocularvision Cyclopeaneye Occlusion Alignmenttask

INTRODUCTION

Viewingby two eyes insteadof one eye providesan extracue for depth perception, but also creates problems forthe perception of direction and distance. One of theproblems is related to the fact that, by definition,direction and distance need references. Concerningbinocular direction, it is generally accepted that we seethe world as if viewed by a single eye. This centre ofbinocular vision, the cyclopean eye (Fig. 1) is definedsuch that its position and visual axis serve as referencesfor binocular visual direction. Hering (1879/1942)formulatedrules for human visualperceptionwhich haverecently been translated into mathematical expressions(van de Grinclet al., 1995).

According to the rules of Hering (1879/1942), allvisual elements viewed by either eye are included in thecyclopean eye and, thus, are perceived in binocularvision. A consequence of these rules is that details ofobjects are visible that cannot be seen by a real eyelocated at the positionof the cyclopeaneye. Recently,wediscoveredthat thispropertygives rise to a paradox in theconcept of the cyclopean eye (Erkelens & van de Grind,1994). The cyclopean as well as the real eyes have thestructure of two-dimensional manifolds on which eachpositionrepresentsa visual direction.The cyclopeaneye,however, contains all the projectionsof visual elementsthat are projected in either the left or the right eye.Generally, one eye will contain a number of projections

* Helmholtz Instituut, Utrecht University, P.O. Box 80000,3508 TAUtrecht, The Netherlands.

~ To whom all correspondence should be addressed [had c.j.erkelens@fys.ruu.nl].

of visual elements that are not visible to the other eye. Inthis case the cyclopean eye must have room for moreprojected elements than either of the eyes. However, thisis impossible because, by definition, the cyclopean eyeand the real eyes have the same dimensions.The paradoxis demonstratedby the cyclopean direction of point P inFig. 1.P would not be visiblefor a real eye at the positionof the cyclopean eye because it is occluded by the bar atF. The virtual cyclopean eye, however, must host theprojection of P because P is visible to the left eye. Weinvestigated the paradox by examining the directions ofmonocularand binocularlines lying in one depthplane inan alignment task (Erkelens & van de Grind, 1994). Ourresultsshowedthat the rules of cyclopeandirectionfailedto predict alignmentwhen one line was presented to oneeye and the other line to two eyes. In such conditionsbinocular alignment was achieved by alignment of thetwo monocular lines presented to a single eye.

From experimentsin which lines are aligned that lie inone depthplane,we cannotdecidewhether the cyclopeanrulesare valid or not for alignmentof two binocular lines.Because binocular alignment is equivalent to monocularalignment for two lines lying in the same depth plane(Erkelens& van de Grind, 1994),in the present study weused binocular alignmentof two binocular lines lying indifferent depth planes as a test for the validity of thecyclopean rules. We measured alignment of binocularlines in two conditions, namely with and withoutmonocularocclusionspresent in the stimulus.The reasonfor making a distinction between these two conditionswas that the cyclopean paradox only exists whenmonocular occlusionsare present. The results show thatthe cyclopean rules of Hering are valid for a stimuluswithout monocular occlusion. In the vicinity of a

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2142 C. J. ERKELENSet al.

A’ A“B” B’1 f

●r 1- 1

A

Left eye

Cyclopean eye

FIGURE 1. Conceptof the cyclopean eye. Top view of a pair of eyesviewingtwo bara located in different fronto-parallelplanes. The smallbar is fixated in F. The large bar is partially occluded from vision bythe small bar. The cyclopean eye specifies the position in space fromwhere the objects are seen. If P were visible to a real eye at thisposition, its visual direction would be to the right of A. According tothe rules of cyclopeanvision,P is perceivedby the cyclopeaneye andits visual direction is to the left of A. Consequently,A andA’ are not

perceived in the same cyclopean direction.

monocularly occluded area, the rules are not valid andbinocular direction of binocularly visible elements isequal to its direction in the eye viewing the monocularlyoccluded area.

METHODS

SubjectsFour subjectsparticipated in the experiments.None of

them showed anyvisualor oculo-motorpathologiesotherthan refraction anomalies. The subjects had normal orcorrected-to-normal vision. They were checked fornormal stereo vision by means of partially decorrelatedrandom-dot test images (Julesz, 1971). Two of thesubjectswere experienced in stereoscopicexperiments.

ApparatusThe stimuliwere generated at a frequencyof 70 Hz by

an HP 750 graphics computer and back-projected on afronto-parallel translucent screen by a projection TV(Barco Data 800). The subjectwas seated about 1.2 m infront of the screen. One image was projected on thescreen after passing through a green filter and wasobserved by the right eye through a green filter. Redfilters were used to make the other image visibleexclusively to the left eye. Between stimuli the screenwas blanked for 2 sec. The subjectswere not restricted intheir head and eye movements.The stimuliwere viewedin a completely dark room.

Procedure

Figure 2 shows a schematic drawing of the stimuli.Figure 2(A) shows the stimulus that was used in thebinocular alignment task without monocular occlusion.The anaglyphic stereogram contained two horizontally

scaled rectangles (60 x 30°) consisting of random dots(dot size: 6 x 6’). The upper and lower rectangles wereoppositely scaled by 2.5, 5 or 7.5Y0.The horizontalpositionsof thevertical lines (6’ x 30°)were scaled in thesame way. In binocularvision, the stereogramcontainedtwo planeswhich were oppositelyslanted to each other inthe horizontaldirection.The upper line could move alongthe slanted upper plane by moving the computer mouse.The lower line was set randomly in the slanted lowerplane at one of 50 horizontalpositionsrunningfrom –25to 25°. Each positionwas presented four times. Subjectswere asked to binocularlyalign the upper with the lowerline.

Figure 2(B) shows the stimulus that was used forbinocular alignment in the neighborhood of a mono-cularly occluded area. The anaglyphic stereogram,containing a circular disk (28° dia) on a squarebackground (60 x 600), was essentially a traditionalJulesz random-dot stereogram (dot size: 6 x 6’). Thismeans that the disk was only visible in binocular vision.In separate sessions we also measured binocular align-ment with this stimulus, but in which we added rims tothe disks that could be seen in monocularvision.The diskwas always presented in front of the surround withdisparities ranging from Oto 2°. The line (width 6’ andheight 50°) could be moved to the left and to the right bymoving the computer mouse. The line lay in the plane ofthe backgroundat all times. Four subjectswere asked toalign thevertical linewith the rim of the disk.The subjectalsomade alignmentswhile viewing the same stereogramin which the lines were oriented horizontally and were

A

B

FIGURE2. Stereogramsusedin binocularalignmenttasks without(A)and with (B) monocular occlusion. Stereogram (A) contained twooppositely slanted planes. The upper lines were movable in thehorizontaldirectionsuch that the line movedalongthe slantedplane inbinocular vision. Subjectswere asked to align the upper line with thepreset lower line. Stereogram(B) containeda circular disk on a squarebackground.In contrast to the different textures in the picture, disk andsquare contained identical random-dot textures in the stimulus. Thediska were presented with and without a monocularly visible rim inseparate sessions.Subjectswere asked to align the movableinterruptedline with the rim of the disk. The stereograms are drawn in this figure

for uncrossedfusion.

BINOCUIAR ALIGNMENTIN DIFFERENTDEPTHPLANES 2143

Lefteye

EA-1——l

Righteye

.-7 -30 0 30-30 0 30

ii0L-(:qu%*:@**\\-2

-30 0 30

Cyclopfxmeye

SubjectCE

-30 0 30 -30 0 30

Cyclopeandirection(deg)

FIGURE3. Differences between settings and positions of the test line as a function of the cyclopean direction of the lines.Differences are computedfor the lines viewedby the left (left panels), and the right eye (middlepanels). The right panels showthe differencesfor the means of left and right eye settings.The dashedlines are predictionsfor a cyclopeaneye located midway

between the eyes.

movable in the vertical direction.Disparitybetween diskand backgroundwas varied randomly.While the subjectsmade the alignmentsthey were free to fixate any part ofthe stimulus.The use of an interruptedline was preferredover a continuous one because it allowed accuratealignment without the loss of fusion when the line wasplaced close to the rim of the disk.

Data analysis

The settings made by the subjects were transformedinto cyclopean directions according to the rules ofcyclopean direction: see rules H1–H5 in van de Grindet al. (1995) or the review by Ono (1991). Theses rulesrequire different transfers for monocularly and binocu-larly visible visual elements. When the test line ismonocularly visible, the rules of cyclopean directionstate that the binocularvisualdirectionof the line relativeto the visual axis of the cyclopean eye is equal to themonocular visual direction relative to the visual axis ofthe eye to which the test line is visible.When the test lineis binocularly fused, the binocular visual direction isequal to the average of the two monocular visualdirections. After transfer, the settings of the subjectswere compared to the computed binocular visualdirectionsof the test lines.

RESULTS

Binocular alignment withoutmonocularocclusions

The subjects found it easy to align the two verticallines. Occasionally,subjects found it difficultto binocu-

larly fuse the lines at large disparities.Trials in which thesubjects did not manage to fuse the lines were excludedfrom further analysis. Figure 3 shows differencesbetween settings of two subjects and directions of thetest line as a function of the cyclopean direction of thetest lines. The results are shown for horizontal scaling of5% between the half-images. The differences betweensettings and preset directions vary more or less linearlywith cyclopean direction, which means that they alsovary about linearly with disparity. The differences areabout linearly related to disparity over the full range ofcyclopean directions in all subjects. This linear relation-ship shows that the centre of binocular direction has afixed position in the head. However, the anatomicalposition of the centre of binocular direction showsindividual differences. Theoretically, slopes of linearregressionsto the data are –0.05 in the left eye and 0.05in the right eye for a centre of binocular directionpositionedexactly halfway between the two eyes. In thiscase, binocularvisualdirectionis equal to the mean of thetwo monocular directions.This means that directions inthe two eyes are equally weighted. The centre ofbinocular direction is located off the median plane ifweighing is unequal. In the extreme case the weightingfactor is zero for one of the eyes, implying that thedirectionalcentre is located at the other eye. If the centreof direction is located at the position of the left eye, theslopesare Oin the left eye and 0.1 in the right eye. If it islocated at the right eye, the slopesare –0.1 in the left eyeand Oin the right eye. Table 1 shows that the centre ofbinoculardirection is located about halfway between the

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TABLE 1. Slopes calculated from linear regressions to the data

Slopes of linear fits Weightingfactors (%)Subjects Left eye Right eye Left eye Right eye

CE -0.046* 0.054” 54CVG

46-0.018” 0.080* 81 19

AM –0.076” 0.026” 26 74RvE -0.009 0.088” 90 10

Asterisks indicate that slopes are significantly different from zero(P< 0.05). Weightingfactors computedfrom the slopes indicatethe contributionsof the two eyes to the binocular direction.

eyes in subject CE. The centre is shifted towards the lefteye in subjectsCVGand RE. In subjectAM it is slightlyshifted towards the right eye.

Binocular alignment in the neighbourhoodof monocularocclusions

We used an interrupted line and a circular disk toexamine alignment near monocular occlusions. Forseveral reasons this stimulus was very suited for thistask. In our red/green stereograms, details that becomemonocularly occluded are associated with a change incolour from yellow to green or red. The gap in the lineprevented the subjects from using a change in colour ofthe line if it was monocularlyoccludedby the disk as aninappropriatecue for alignment.The circular shapeof thedisk allowedthe gap in the line to be smallwhich enabledaccuratealignment.A furtheradvantageousfeatureof thegap in the line was that relative disparitiesbetween lineand disk remained spatiallyseparatedfrom each other. Inthis way the disparity gradient remained within fusibIeranges. Indeed, the subjects found the alignment taskvery easy and they hardlyever lost fusion.Figure4 showsmean settingsas a function of disparitybetween disk andline for binocular alignment at the different sides of thedisk.At zero disparity,alignmentis in agreementwith thecyclopean direction of the rim of the disk, As disparityincreases, the cyclopean directionof the rim remains thesame. However, the left half-image of the disk (viewedby the left eye) shifts to the right and the righthalf-image(viewed by the right eye) shifts to the left by equalamounts. These shifts are equal to half of the disparity.The left panel in Fig. 4 shows that, at the left side of the

Disparity(deg)

FIGURE4. Differences in cyclopean directions of line and disk as atimction of disparity between line and disk. Means and SD are shown

for subject AM.

disk, the directions indicated by the subject shift to therightby half of the disparity.This means that the subjectsaligned the line and the disk as viewed by the left eye.This result was the same in the four subjects. Similarresultswere obtainedon the right side of the disk. On thisside, the “directionsindicated by the subject shift to theleft by half of the disparity (Fig. 4, right panel). Thisshows that the subjects aligned the line and the disk asviewed by the right eye. Settings of alignment wereindependentof disparityon the upper and lower sides ofthe disk. Alignment qualitatively followed the samepattern if the monocularly visible rims of the disk wereremoved.A noticeabledifferencewas that alignmentwasmore accurate in the presence of the monocular rims.

DISCUSSION

Binocular visual direction

We measured the binocularalignmentof lines lying indifferent depth planes in a random-dot stimulus withoutmonocular occlusion.The results of this experiment arein agreement with the prevailing notion of binocularvisual direction (One, 1991). The binocular visualdirections of objects are judged from a fixed centre inthe head, called the cyclopean eye. Objects havebinocular visual directions that are averages of the twomonocular visual directions. Experiments in which thecontrast of the monocular stimuli were manipulatedshowed that the averages are weighted by the sensoryinputs from each eye (Mansfield & Legge, 1995). Ourresults show that the weighting factors for the averagingof monoculardirectionscan be considerablydifferent inindividualsubjects. Individualdifferencesin the locationof the centre of binocular projection (or in left/rightweights for determining binocular direction) have beenpreviously reported in studies of Sheedy and Fry (1979)and Porac and Coren (1986).

Recently, we examined binocular visual direction inmonocularly occluded areas (Erkelens & van de Grind,1994).We found that if one line is presented to one eyeand the other line to two eyes, binocular alignment isachieved by alignment of the two monocular linespresented to the same eye. This result is not predictedby the rules of cyclopean direction. In the present study,we measured the binocular alignment of binocularlyvisible lines lying in different depth planes in theneighborhood of monocularly occluded areas. Againwe find that the cyclopeanroles do not predict alignmentand that alignment is achieved by alignment of mono-cular lines presented to the same eye. In each particulardirectiononly one of the eyes can be used for alignment,because the two eyes give different results.Which of theeyes is used for alignment is not a matter of choice butseems to be related to the structureof the stimulus.In ourstimulus,the left eye was used for alignment on the leftside of the disk, whereas the right eye was used on theright side. Since the subjects were free to fixate thestimulus wherever they liked, it is most likely that theprocess, determiningwhich eye is used for alignment, is

BINOCULARALIGNMENTIN DIFFERENTDEPTHPLANES 2145

A

B

c

FIGURE 5. Random-dotstereogram demonstratingthe incompatibilityof binocularly perceived shape and direction from a-.single point of view. Viewingof the left two patterns gives the appropriatestereogramfor “uncrossed” fusion, for “crossed”

fusion one needs to view the right two patterns. (A), (B) and (C) are explained in the text.

related to aspects of the stimulus and not to specificretinal locations. Alignment in our experiment followsthe rule that the eye which views the monocularlyoccluded area is used for alignment in that .neighbour-hood. This rule of alignment was followed by all oursubjects. It is remarkable to see that near monocularlyoccludedareas, all the subjectsalignedthe lines in a verysimilar way, whereas alignment in stimuli withoutmonocular occlusions showed considerable individualdifferences.

Two hypothesesabout binocularvisual direction

In the experiments,the left eye was used for alignmenton the left side of the occludingdiskand the righteye wasused on the right side. For the two bars shown in Fig. 1this rule for alignmentimpliesthatA” insteadofA’ is seenin same direction asA. Similarly,B“ is alignedwith B. Ifperception of three-dimensional space is veridical, theobserved alignments imply that viewing from a singlepoint of view, i.e. the cyclopean concept, is notapplicable to human binocularvision. It is only possibleto view the world as if from a singleposition in the headif binocular space perception is not veridical. If the latteris the case,A“ is indeed aligned withA (Fig. 1), butA“ isperceived at the positionofA’. Similarly,B“ is then seenat the positionof B’. The consequenceof such a distortedspace perception must be that the length of linepieceA“B” is perceived too long relative to the length ofIinepiece AB. This relative lengthening of occludedlinepieces can only occur in the horizontal direction,because alignment is in agreement with the cyclopeanrules in the vertical direction (see the results in Fig. 4).The consequence of horizontal lengthening of partiallyoccluded surfaces must be that their shape will beperceived differentlyfrom the shape of the same surfaceif it is not occluded. Very recently, Ohtsuka (1995) usedthe horizontal lengthening of occluded linepieces to

explain the perceived misalignmentof the oblique line inthe Poggendorff illusion.

Our experiments do not tell us whether the hypothe-tical broadening of occluded surfaces really occurs andwhether we misperceive the directions of monocularlyoccluded points or not. However, based on the formerreasoning we can formulate two testable, alternativehypotheses about the concept of binocular visualdirection: (1) the cyclopean eye has a fixed position inthe head and binocular space perception is distortednearmonocularly occluded areas; or (2) binocular spaceperception near monocularlyoccluded areas is veridicaland the cyclopean eye does not have a fixed position inthe head, but is located between the eyes for certainvisual directions and in one of the eyes for otherdirections. The stereogram of Fig. 5 demonstrates thatthe second hypothesis is more likely to be true than thefirst one.

The top (A) and bottom (C) parts of the stereogramprovide equal images for the two eyes. In binocularvision, theseparts of the stereogramare seen in one depthplane. The middle part of the stereogram (B) generatestwo depth planes. In addition to the random dots, thestereogram contains a number of monocularly identifi-able line drawings.The large squares in (A), (B) and (C)have the same size in monocular vision. The largesquares in (B) are partially occluded from vision. Inbinocularvision, the non-occludedand partiallyoccludedlarge squares all have the same shape, namely the shapeof a square.The small squares in (B) mark the rim of thefrontal depth plane. The small squares in (A) have thesame size as those of (B) when viewed monocularly.Therectangles at (C) have smaller widths than the smallsquares in (A) and (B). In binocular vision, the widthsbetween the square in the foreground and the square inthe background in (B) are equal to those between therectangle and the large square in (C). The horizontal

2146 C. J. ERKELENSet al.

A

B

c

FIGURE6. Random-dotstereogramdemonstratingthe influenceof monocularocclusionson flankingvertical lines. Viewingofthe left two patterns gives the appropriatestereogramfor “uncrossed” fusion, for “crossed” fusionone needs to view the right

two patterns. (A), (B) and (C) are explained in the text.

widths between the small and large squares in (A) arevery different from those in (B). It is curious that we cansee a square in front of another square of which the non-occludedareas are differentfor horizontaland for verticaldirections.

The stereogram of Fig. 5 shows that the square in theforegroundplane (B) occludesan area in the backgroundplane that has the size of a rectangle (C). At the sametime, squares in the background have the same shapeirrespectiveof whether they are partiallyoccludedor not.The difference in perceived shape of an occluding andoccluded surface is not possible if the world is viewedfrom one centre. Apparently, we have more than onecentre of binocular visual direction. The alignments onthe left and right side of the disk (Fig. 4) show that thetwo eyes serve as centres near monocular occlusions,each of them for differentplaces in the visual scene. Ourexperiment, in which we measured binocular alignmentwithoutmonocularocclusions,showsthatwe have a thirdcentre located at some place between the eyes fromwhich binocular visual direction is judged if monocularocclusions are not present. We support the view of Onoand Barbeito (1982) that the cyclopean eye serves as thecentre of visual directions in these stimulus conditions.Near occlusions, however, the sighting eye, or moreprecisely the eye viewing the monocular occlusion,serves as the centre.

Locally dominant centres of binoculardirectionMonocularocclusionsare an interestingobjectof study

in relation to the binocular perception of depth anddirection. The depth of monocular occlusions,called daVinci stereopsis, has been extensively studied byNakayama and his co-workers (Nakayama & Shimojo,1990;Shimojo& Nakayama, 1990,1994;Takeichiet al.,1992; Anderson & Nakayama, 1994). An interestingfinding was that monocular occlusions are localised indepth despite the lack of explicit disparity information(Nakayama & Shimojo, 1990). This suggests that depth

information is transferred from the neighboring bino-cular regions to the monocularocclusion.Takeichi et al.(1992) found a similar depth-spreading effect betweendisparity stimuli and illusoryoccluded surfaces. Anotherfinding of Shimojo and Nakayama (1990) was that amonocularlyoccluded region can only escape binocularrivalry if it is ecologically valid. The authors concludefrom this result that binocular rivalry is criticallydependenton which eye receives the unpaired stimuli inrelation to local depth signals. This conclusion isquestionable because it is based on an inappropriatedistinction between valid and invalid monocular occlu-sions. The point is that invalid monocular occlusionsdoneither occur in natural scenes nor can be simulated oncomputer screens. The reason is that monocular occlu-sions determinethe amount and sign of disparity of theirneighboring regions.Changingthe monocularocclusionfrom one eye to the other eye, but leaving the disparitybetween the neighboring regions unchanged is notpossible. In the case of ‘valid’monocularocclusions,themonocularocclusionhas no competition in the other eye(the neighbors of the monocular occlusion are neigh-bours~ each other in the other eye). This means thatbinocularrivalry is not possible(it takes two to tango). Inthe case of ‘invalid’ monocular occlusions, the mono-cular occlusioncompeteswith anotherunpaired region inthe other eye leading to binocular rivalry.

Our experiment and demonstration show that the eyeviewing a monocular occlusion serves as the centre forbinocular alignment, not only inside, but also in theneighbourhood of the monocular occlusion. Figure 6demonstrates that alignment near monocular occlusionsis associated with local ocular dominance. By localocular dominancewe mean that in a limitedvisual regionthe visual stimulusof one eye dominates the stimulusofthe other eye. The stereogramof Fig. 6 shows three equalsquares hovering above the background. Square (B) isflankedby two binocularly visible lines which are lyingin the background.In each of the two half-imagesof the

BINOCULARALIGNMENTIN DIFFERENTDEPTHPLANES 2147

stereogram, one line in (B) neighbors a monocularlyoccludedregion and the other one does not. Square (A) isonly flanked by the lines that border on monocularocclusions, the other lines are replaced by random dots.In binocular vision, the lines are stable and perceived inthe same direction and depth plane as the lines of (B).Square (C) is flankedby the lines that do not neighbouramonocular occlusion and the other lines are replaced byrandom dots. In binocularvision, the lines rival with therandom dots. In a number of observersthe lines are evencompletelysuppressed.This stereogramshows that, nearocclusions,not only the directions of visual elements inone eye are outweighted by those of visual elements inthe other eye, but also the visual elements themselves.This associatedbehavioursuggestsa connectionbetweenthe processes that induce binocular direction andbinocular rivalry.

The stereogramsof Figs 5 and 6 show that monocularocclusions play a key role in binocular visual directionand in binocular rivalry. As far as we know, thisconnection between binocular visual direction andbinocular rivalry has not been proposed in the literature.We suggest that both binocular visual direction andbinocular rivalry are controlled by a process of localsuppression. This process may change the weightingfactors of sensory signals from the two eyes, in this wayshifting the centre of binocular visual direction(Mansfield & Legge, 1995). The powerful influence ofmonocular occlusions on binocular visual direction andbinocular rivalry suggests that suppressionis not limitedto corresponding positions or features, but laterallyspreads to neighboring regions.

Consequencesfor binocularspaceperceptionFor more than a hundred years, many conclusions,

ideas and mc)delsrelating to binocular vision have beenbased on the conceptof the cyclopeaneye. The literaturedealing with this concept needs revision. For instance,much work has been concentrated on the geometry ofbinocular visual space. Many studies show that visualspace is not Euclidean (for a review see Foley, 1991).Luneburg (1947, 1950) and Blank (1953) proposed atheory in which they predict that visual space isRiemannian. Later experiments show that visual spaceis probablynot Riemannianeither (Foley, 1972;Indow&Watanabe, 1.984). Until now, all results have beenobtained from experiments in which the physical spacewas almost empty. The present results show that a spacefilled with occluding objects, the normal condition indaylight vision, generates a different visual space. Animportant implication of locally dominant centres of

direction is that visual directionsdepend not only on thepositionsof objects in physicalspace relative to the head,but they also depend on which eye is serving as the localcentre of direction. The conclusion is that binocularvisual space cannot be fully described by globalgeometry.

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